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GeoGebra Functions


Index

Activity Topic Page 1 Introduction GeoGebra Functions 3

2 To draw the function f(x)=x2 4

3 To change the colour, etc. of a graph of a function 5

4 To draw the graph of the quadratic function ( ) 5

5 To animate the above graph of the function using the slider a 7

6 To export an animated slide to PowerPoint 7

7 To investigate a quadratic of the form f(x)=a*(x+b)^2+c 8

8 To draw a section of a function 9

9 To show a function appearing bit by bit 9

10 Find where the graph of a function and a line meet 10

11 To draw the graph of a cubic function using sliders 11

12 Highlighting where one function is above another 11

13 Turning Points, Roots, Derivatives, Second Derivatives and Integrals

13

14 To demonstrate derivative and slope 14

15 Function Inspector Tool 16

16 Using the Function Inspector to demonstrate the derivative of a quadratic is linear

21

17 To transfer points from the Function Inspector to the Spreadsheet View

23

18 To demonstrate integration and area 24

19 Integration and the Trapezoidal rule 25

20 To find the area between the graph of a function and the x axis, where there is a section below and a section above the x axis

26

21 To demonstrate the average value of a function between 2 points

26

22 To draw the absolute value function 29

23 To draw ex, ln(x) and look at the (inverse) relationship between them

30

24 To change the x axis to radian measure and draw a trigonometric function

31

25 To change the x axis to degree measure and draw a trigonometric function

32

26 To plot data in the Spreadsheet View in the Graphics View 33

27 To fit a graph to a list of points that are shown on the Graphics View

33

Appendix A A1: How to save a file as a GeoGebra file A2: How to save a file as an interactive web page

34

Appendix B How to use the virtual Keyboard in GeoGebra

37


Activity 1: Introduction GeoGebra Functions

To download GeoGebra go to www.geogebra.org and the following window will appear.

Click Free Software and the following window will appear.

Click the platform suitable for your computer and follow the indicated steps to download

GeoGebra.

Note: In order to run GeoGebra it is essential that an up to date version of Java is available

on your machine. This package is available at free at http://www.java.com/en/ .

In GeoGebra there are three Views, Algebra, Graphics and Spreadsheet. The bar containing

File, Edit etc. is known as the Menu bar.

http://www.geogebra.org/

http://www.java.com/en/


Note: When you click on the Graphics View the following toolbar appears:

and when you click on the Spreadsheet View the following toolbar appears:

Note: If the Spreadsheet View is not visible go to View and click Spreadsheet and if the

Graphics View is not visible go to View and choose Graphics.

Note: When drawing a function use ( ) rather than because when is used

some of the commands from the Input Bar do not work for the function.

Activity 2: To draw the function f(x)=x2

1. Open GeoGebra.

2. In the Input Bar type ( ) and press Enter on the keyboard. Pressing the Shift

key and the 6 key simultaneously on the keyboard gets ^.

Graphics

Spreadsheet Algebra

Input Bar


Activity 3: To change the colour, etc. of a graph of a function

1. Click on the graph of the function, right click and choose Object Properties. A new

dialogue box appears.

2. With the Properties tool open choose the Colour tab to change the colour and

choose the Style tab to change the style of the line.

3. With the Basic tab open, click the Show Label button and choose Name and Value

from the drop down menu to enable the name of the graph of the function and its

equation both to be shown. Click at the top of this window.

Activity 4: To draw the graph of the quadratic function f(x) = ax2+bx+c

1. Go to File and choose New Window.

2. Select the Slider tool . Click on the Graphics View and create a slider called a

with min: = -5, max: = 5 and Increment: = 1.


3. Select the Slider tool . Click on the Graphics View and create a slider called b

with min: = -5, max: =5 and Increment: = 1.

4. Select the Slider tool . Click on the Graphics View and create a slider called c


5. In the Input Bar type f(x)=a x^2+b x+c Note: The spaces can be replaced by an *.

6. Move the sliders and see how the graph of the function changes.

Explorations:

What is the effect of changing the “a” slider?

What is the effect of changing the “b” slider?

What is the effect of changing the “c” slider?

7. Open the Algebra View if it is not already open (Go to View and click Algebra) and

watch the value for f(x) change as the sliders change.

8. In order to see the value of the function in the Graphics View, right click the graph of

the function and select Object Properties. With the Properties tool clicked and

the Basic tab open click the Show Label button and select Name and Value from the

drop down menu. Click at the top of the dialogue box.

Alternatively to see the equations change in the Graphics View:

1. Select the Text tool and click on the Graphics View at the position you want

the formula for the graph of the function to appear. A new dialogue box appears.

2. Type the following text “f(x) =” in the Edit section and select f from the drop down

menu under Objects and click OK.


Activity 5: To animate the above graph of the function using the slider a

1. Right click on slider a and choose Animation On.

2. A new icon will appear at the bottom left hand corner of the Graphics View. Click

on this icon to stop and start the animation.

Activity 6: To export an animated slide to PowerPoint

1. Make the GeoGebra file using a slider as normal. For example the slider a and the

function f(x) = a x+1.

2. Save the file, by going to File and Save As.

3. Go to File, Export and Graphics View as an Animated GIF. A new window appears.

4. Choose the slider you require to be animated by following the drop down menu

beside Slider. To enable the animation to continue infinitely click the As Loop box.

Choose the Time between Frames and click Export.

Note: The exporting of this gif can be slow.

5. Open PowerPoint and insert your picture in the usual way depending on the version

of PowerPoint you have.

6. View the slide show.


Activity 7: To investigate a quadratic of the form f(x) = a*(x+b)^2+c



with min: = -5, max: = 5 and Increment: = 1.

3. Select the Slider tool . Click on the Graphics View and create a slider called b


4. Select the Slider tool . Click on the Graphics View and create a slider called c


5. In the Input Bar type f(x)=a(x+b)^2+c Press Enter on the keyboard.

6. Move the sliders and see how the graph of the function changes.

Explorations:

What is the effect of changing the “a” slider?

What is the effect of changing the “b” slider?

What is the effect of changing the “c” slider?


Activity 8: To draw a section of a function


2. In the Input Bar type Function[x^2, 0,4].

3. Press Enter on the keyboard.

Activity 9: To show a function appearing bit by bit


2. In the Input Bar type the f(x)=x^2-4 x-3 to create the function f(x) = x2-4x-3.

3. Hide the function f. (Right click the function f and click Show Function.)

4. In the Input Bar type start=–1. Press Enter on the keyboard.

In the Input Bar type end=5. Press Enter on the keyboard.

This is defining two numbers. The numbers could be called anything e.g. a and b, but

start and end make more sense later.



with min: = 0, max: = 100 and Increment: = 1. Click OK to close the slider dialogue

box.

6. In the Input Bar type function[f,start,start+(end-start)/100 a]. Press Enter on the

keyboard. Note: The space between 100 and a. This draws a new function which is

the old function divided into 100 parts and can be revealed one part at a time.

function[f,start,start+(end-start)/100 a]

7. Adjust the value of the slider a to reveal the function in sections.

8. Right click on the function g and choose Object Properties and with the colour tab

open change the colour of function g.

Note 1: You can change the values of the start and end numbers later by double-clicking

on them in the Algebra View or by typing e.g. start=2 into the Input Bar.

Note 2: slowplot[f] will do the same as the above, except the domain will be the visible

portion of the Graphics View.

Activity 10: Find where a graph of a function and a line meet


2. Draw the quadratic function f(x) for example f(x)=x2+3x+2 and the line g(x) for

example g(x)=2x+3.

The part of the function to

the right of start is being

divided into 100 pieces.

If your slider a=23 then 23 of the 100 pieces of

the function to the right of start will be shown


3. Select the Intersect Two Objects tool . Click on the graph of the function and

the graph of the line to find their points of intersection.

Point(s) will appear at the intersection of the line and the graph of the function.

Another method is in the Input Bar type Intersect[f,g] and press Enter.

Activity 11: To draw the graph a cubic function using sliders


2. Select the Slider tool . Click on the Graphics View and create sliders a, b, c and

d.

3. In the Input Bar type f(x)=ax^3+bx^2+cx+d. Press Enter on the keyboard.

Note: If you already know the equation of the cubic for example f(x)=2x3+3x2+1x+1,

just type f(x)= 2x3+3x2+1x+1 into the Input Bar and press Enter on the keyboard. You

can also enter equations like f(x)=(x-2)(x+2)(x-1) in the Input Bar.

Activity 12: Highlighting where one function is above another function



2. In the Input Bar type f(x)= x^3+2x^2–3x–1 to create the function

f(x) = x³+ 2x²–3x–1.

3. In the Input Bar type g(x)=x+4 to create the function g(x)=x+4.

4. Type Intersect[f,g] in the Input Bar and press Enter on the keyboard which creates

three points A, B and C. Alternatively use the Intersect Two Objects tool .

5. Type Function[f,x(A),x(B)] in the Input Bar.

A function called h(x) = x3+2x2-3x-1 now appears in the Dependant Objects list in the

Algebra View. Right click on h(x) in the Algebra View and change the colour of this to

blue and its line style to be thicker.

6. Input Function[f,x(C), ∞]

To type the symbol click the on the right hand side of the Input Bar and choose

.

A function called p(x) = x3+2x2-3x-1 now appears in the Dependant Objects list in the

Algebra View. Right click on h(x) in the Algebra View and change its colour and line

thickness to that of h(x).

7. Create a checkbox to show/hide h(x) and p(x). To do this select the Check Box to

Show /Hide Object tool . Click on the Graphics View and when the dialogue

box appears type in some suitable text and then from the drop down menu under

Select objects in construction or choose from list select h(x). Repeat this procedure

for a Tick box to show/hide p(x).


Activity 13: Turning Points, Roots, Derivatives, Second Derivatives and

Integrals


2. In the Input Bar type f(x) = x^3+x^2-3x-2.

3. To find the turning points A and B in the Input Bar type TurningPoint[f].

Note 1: If the language is set to English (US) then you will have to use Extremum[f].

Note 2: By right clicking the points A and B and choosing Object Properties you can

change the names of the Turning Points to something more suitable such as turn1

and turn2.

4. To draw the points where the function f crosses the axis or the roots of the

function f in the Input Bar type root[f].

5. To draw the derivative of f in the Input Bar type derivative[f]. This creates a new

function called f’(x).

Note: An alternative here would have been to type f’(x) into the Input Bar.


6. If you wanted to compare f’(x) with f(x) then type root[f’]in the Input Bar and look at

the x values of the roots of f’(x) and the turning points of f(x).

7. To graph the second derivative of f in the Input Bar type derivative[f’]. This creates a

new function called f’’(x).

Note: An alternative here would have been to type f’’(x) into the Input Bar.

8. In the Input Bar type Integral[f’’]. This draws the integral of f’’(x) and names it g(x).

It is very similar to the graph of f’(x). f’(x)=3x2+2x-3 and g(x)= 3x2+2x.

Note 1: The integral[] command doesn’t add in the “c” bit of the integration.

Note 2: Use of colour and show/hide checkboxes can make things a little clearer

here.

Activity 14: To demonstrate derivative and slope


2. Select the slider tool . Click on the Graphics View and create a slider called a.

3. Select the slider tool . Click on the Graphics View and create a slider called c.

Note: You need to put c for the name of the slider.

4. By typing f(x) =a x^2+x+c in the Input Bar create a quadratic. Note the space

between a and x. This prevents the function being (ax)2+x+c.

5. Right click on the graph of the function and go to Object Properties and change the

colour and the style of the graph of the function as required.

6. Select the Point on Object tool . Click on f and create a point A on f(x).

7. In the Input Bar type Tangent[A, f].


8. To find the slope of the tangent at the point A. Select the Slope Tool . Click on

the Tangent.

9. To create a point B, type B=(x(A), m) in the Input Bar and press Enter on the

Keyboard.

10. Right click on the point B and click the Trace on. Move the point A.

11. In the Input Bar type f’(x). Note the relationship between the trace as you drag point

A and the graph of the function of f’(x).


12. Select the Check Box to Show / Hide Object Tool . Click on the Graphics View and

Complete the new dialogue box that appears as follows:

13. Click Apply.

14. Move the point A.

Note 1: The route of the trace of the point B follows the f’(x) which is the derivative

of f(x).

Note 2: Clicking View and Refresh Views will clear the trace.

Activity 15: Function Inspector Tool Note when the keyboard is mentioned in these notes you can use the keyboard on your machine

or the virtual keyboard supplied with GeoGebra. See Appendix B of these notes on how to access

this keyboard.

1. Draw a function ( ) ( )( )( ) by typing

( ) ( )( )( ) in the Input Bar at the bottom of the screen and press

Enter on the keyboard.

2. In the third set of tools from the right of the toolbar click on the Function Inspector

tool .


3. Select the function in the Graphics View that you want details of. In this case click on

( ) ( )( )( )

4. A new window appears. (Watch this may contain settings from your previous use of

GeoGebra.)

5. With the Interval tab open type the interval you want to examine for example

.

N.B. You must press the Enter button on your keyboard after each number for it to take

effect.

6. The interval is marked on the curve in red and the information like Min, Max, Root,

Integral and Area are now given for that interval in the table.


Note: The Max and the Min that is given in this table is the maximum value and

minimum value of the function in the interval that the Function Inspector is

investigating.

7. The Mean is the “Average value” of the function. It is the area underneath the curve

divided by the length of the interval.

8. If you do not wish the interval to be shown on the Graphics View, you can right click on

this section of the function and deselect Show Object.

9. Open the Points Tab.

Note: If the point shown on the table above is not visible on the Graphics View replace

it with a value that is visible.

10. Click on the Show table of points button . Notice the and ( ) values appear on

the table.


11. Note the point that is highlighted in the table will have a red dot on the graph.

12. To change the Step to 1. Delete the current value using the Backspace button on

your keyboard, press 1 and Enter on the keyboard.

13. To change the values in the table, move the red point on the Graphics View that now

appears on the curve until the required points appear in the table. Four points to the

right and four to the left of the red point will appear on the table.

14. If you just require integer values for the co-ordinate go to Options, Point Capturing

and choose Fixed to Grid. Move the red dot on the Graphics View. You can also

double click on the coordinate of the point highlighted in the table and type in the

number you want.

15. By clicking on Show tangent line button tool you can show the tangent at

the point highlighted in the table.

16. By clicking on the Show x, y position lines button tool you can show the

position of the point highlighted in the table.


17. To add the derivative, second derivative, difference or curvature to the table click on

the button and choose Derivative, 2nd Derivative, Difference or Curvature.

18. To remove the right column on the table click on the button.

19. To add other columns to the table re-click the button as often as required.


N.B. Work done using the Function Inspector cannot be saved, it must be done in class.

Activity 16: The demonstrate the derivative of a quadratic is linear using the

Function Inspector tool

1. Go to File and New window to start a new file.

2. Select the Slider tool .

3. Click on the screen and a new window appears.

4. Change the Min to , Max to 10 and Increment to 1. Click Apply. This creates a slider

called a.

5. Repeat steps 2, 3 and 4 above to make a slider called b, with the Min equal to , Max

equal to 10 and Increment equal to 1.

6. Repeat steps 2, 3 and 4 above to make a slider called c, with the Min equal to , Max

equal to 10 and Increment equal to 1.

7. In the Input Bar type f(x)=a x^2+b x+c and press Enter on the keyboard. Note the space

between the a and the x and the b and the x.

8. In the Input Bar type f’(x). This will enable the first derivative of the function f(x) to

appear on screen. (Provided your original function is called f.)


9. Go to Function Inspector and click on the derivative of the function.

10. Open the Points tab and click on the Show table of Points button .

11. Click on the + button and choose Difference. Notice the values for the differences

are all the same.

12. Move the sliders and while the values for the Difference changes depending on the

values of your sliders you will notice the values for the Difference all stay the same

for all values of for a particular set of values for the sliders.


Activity 17: To transfer points from the Function Inspector to the

Spreadsheet View 1. If the Spreadsheet View is not already in view, go to View and choose Spreadsheet.

2. Highlight the data in the Function Inspector by dragging the cursor over them. Press

Control and C on your keyboard.

3. Click on the Spreadsheet where you wish to insert the data and press Control and V on

your keyboard.


Activity 18: To demonstrate integration and area


2. By typing f(x)=x^2+3 draw the function f(x) = x2+3.

3. Select the Slider tool . Click on the Graphics View and create a slider called a.

4. Select the Slider tool . Click on the Graphics View and create a slider called b.

5. Move the slider b to a value greater than the value of the slider a.

6. In the Input Bar type (a, 0) and press return. The point A(a,0) will now be marked on

the diagram.

7. In the Input Bar type (b, 0) and press return. The point B(b,0) will now be marked on

the diagram.

8. In the Input Bar choose the command Integral from the drop down menu at the

extreme right of the Input Bar and inside the brackets type f,x(A),x(B).

Note: x(A) takes the x value of the point A.

9. Move the points A and B to see how the area c changes.


Activity 19: Integration and the Trapezoidal Rule


2. Create and find the integral of the function as in the activity above. (If you still have

the last activity you can use it.)

3. Select the Slider tool . Click on the Graphics View and create a slider called n

with min:=1, max: =50 and Increment:= 1.

4. In the Input Bar type b= TrapeziumSum[ f,x(A),x(B),n].

5. Move the slider n and as n gets larger check the relationship between c (Area got by

Integration) and d (Area got by Trapezoidal Rule).

Note: In this example the value for the Trapezoidal area between A and B when n =

50will appear to have the same value as c, the integral value between A and B because

the numbers are rounded to two decimal places.


Activity 20: To find the area between the graph of a function and the x axis,

where there is a section below and a section above the x axis


2. Draw a graph of a function for example f(x) = x3-x by typing f(x)=x^3-x in the Input

Bar.

3. Select the Intersect Two Objects tool . Click on the graph of the function and

the x axis to find the points of intersection A, B and C of the graph of the function

and the x axis.

4. In the Input Bar type a=Integral[f,x(A),x(B)].

5. In the Input Bar type b=Integral[f,x(C),x(B)]. Note. The order of A, B and C.

6. In the Input Bar type t=a+b to get the total area between the graph of the function

and the x axis.

Activity 21: To demonstrate the average value of a function between 2

points


(For example f(x) = x2 between x=1 and x = 3)


2. By typing f(x)=x2 in the Input Bar, draw a graph of a function for example f(x) = x2.

3. Select the Move tool . Click on the x axis and drag to the right to extend the x

axis.

4. Create point A at (1,0) and point B at (3,0).

5. Select the Perpendicular Line tool . Click on point A and the x axis to draw a

line a perpendicular to the x axis at A.

6. Select the Perpendicular Line tool . Click on point B and the x axis to draw a line

b perpendicular to the x axis at B.

7. Select the Intersect Two Objects tool . Click on the line a and the function f to

find the point of intersection C of the line a and the function f.

8. Select the Intersect Two Objects tool . Click on the line b and the function f to

find the point of intersection D of the line b and the function f.

9. Select the Point on Object tool . Click on the x axis between the points A and B

to create a point E.

10. Select the Perpendicular Line tool . Click on the x axis and the point E to draw

a line c perpendicular to the x axis at the point E.


11. Select the Intersect Two Objects tool . Click on the line c and the function f to

find the point of intersection F of line c and the function f.

12. Select the Parallel Line tool . Click on the point F and the x axis to draw a line d

through the point F parallel to the x axis.

13. Select the Intersect Two Objects tool . Click on the line d and the line a to find

the point of intersection G of the line d and the line a.

14. Select the Intersect Two Objects tool . Click on the line d and the line b to find

the point of intersection H of the line d and the line b.

15. Right click on the lines a, b, c and d and deselect Show Object.

16. Select the Polygon tool .Click on the points A, B, H, G and A to draw the

polygon ABHG.

17. Right click on the polygon ABHG, choose Object Properties and with the Colour tab

open choose a bright pink colour and with the Colour tab open choose an Opacity of

50.

18. In the Input Bar type i= Integral[f,x(A),x(B)]. This will return the value of the area

between the graph of the function and the x axis in the interval x=1 to x=3.

19. Right click the Integral in the Algebra View and with the Colour tab open choose a

bright blue and Opacity of 50.

20. Move the point E along the x axis until the area of the polygon ABHG is equal to the

value of the Integral i. At this point the area of the pink section of the polygon will

equal the area of the blue section on the diagram. Hence the y value of point F is the

average value of the function between A and B.


21. Checkboxes to Show / Hide Objects and Text boxes may be added.

Activity 22: To draw the absolute value function


2. In the Input Bar type for example f(x)=abs(x-3) to draw the absolute value of the

function x-3.


Activity 23: To draw ex, ln(x) and look at the (inverse) relationship between

them


2. In the Input Bar type f(x)=exp(x)

3. In the Input Bar type g(x)=ln(x)

4. In the Input Bar type y=x

5. Select the Check box to Show / Hide Objects tool . Insert the caption ln(x)

and in the drop down menu under Select objects in construction or choose from list

choose the function g.

6. Select the Point on Object tool . Click on the exp(x) function to create a point A

on it.

7. Select the Reflect Object in Line tool . Click on the point A and then the line y=x

to reflect the point A in the line y=x. This creates a point A’.

Note: Reflecting in the line y = x produces the inverse function.

8. Right click on A’ and click Trace On.

9. Select the Move tool . Click on the point A and move it along the graph of the

function f(x) = exp(x) (slowly is best). A’ should trace out a path.


10. Click the checkbox you created earlier and compare the path of A’ to the ln(x)

function.

Note: Clicking View in the Menu and Refresh Views from the drop down menu will

clear the trace. Ctrl+F also works to clear the trace.

Activity 24: To change the x axis to radian measure and draw a

trigonometric function

1. Right click on the Graphics View and choose Graphics.

2. With the Preferences – Graphics button clicked, Click on the xAxis tab of the

new dialogue box.

3. Click the arrow beside the Distance box and in the drop down menu beside Distance

choose π or π/2. Click Close.


4. In the Input bar type for example and press Enter on the

keyboard.

Activity 25: To change the x axis to degree measure and draw a trigonometric

function

1. Right click on the Graphics View and choose Graphics.

2. With the Preferences – Graphics button clicked, click on the x Axis tab of the new

dialogue box.

3. Click the Distance box if not already clicked and type for example 10 if you want it to go

up in tens. Using the drop down menu beside Unit choose the degree symbol. Click the

X at the top of the dialogue box.

4. In the Input Bar type . Click on the α at the right hand side of the

Input Bar to get the degree sign.

5. If necessary use the Move Graphics tool to decrease the scale of the x axis.


Activity 26: To plot data in the Spreadsheet View in the Graphics View

1. Highlight two columns of data in the Spreadsheet, right click, chose Create and List of

points.

Activity 27: To fit a graph to a list of points that are shown on the Graphics

View

1. In the Input Bar type Fitpoly[list1,2], if the list is list1 and you require a curve of

degree 2 for example.


Appendix A

A1: How to save a file as a GeoGebra file

1. Go to the Menu and choose Save as from the drop down menu. A new dialogue box

appears.

2. For Save in, navigate to the folder that you wish the file to be saved in.

3. For File name type the name you wish to give your file.

4. Click Save. A file with the .ggb extension is created. This extension identifies the file

as a GeoGebra file and it can be opened again and adjustments made using the

GeoGebra application.

A2: How to save a file as an interactive web page

1. First save your file as a GeoGebra file.

2. Go to File, from the drop down menu select Export and from the further menu select

the option Export Dynamic Worksheet as Webpage (html). A new dialogue box

appears.


3. Click on the Export as Webpage tab.

4. Add Title, Author, etc. Then click on the Advanced tab and a new dialogue box

appears.


It is recommended that one clicks the Show Icon to reset construction button.

5. Click Export.

6. Navigate to the folder you want to save your file in. Give it the name of your choice.

Note: This file will have a .html extension and will be viewed using an Internet

browser such as Internet Explorer 8.

7. Click Save and your default Internet browser will open.

8. When the webpage comes up you the following message will normally appear at the

top of the screen.

9. Click Allow Blocked Content.


Appendix B: How to use the Virtual Keyboard in GeoGebra

1. Go to View and choose Keyboard.

2. A virtual keyboard appears.

3. If the keyboard below does not appear click on the key on the virtual keyboard

that appeared.

4. Then click on the key on the virtual keyboard to get the Maths keyboard.

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